Jari Taskinen

COMPOSITION OPERATORS BETWEEN WEIGHTED BANACH SPACES OF ANALYTIC FUNCTIONS

We characterize those analytic self-maps ’ of the unit disc which generate bounded or compact composition operators C’ between given weighted Banach spaces H 1 v or H 0 v of analytic functions with the weighted sup-norms. We characterize also those composition operators which are bounded or compact with respect to all reasonable weights v.

On the spectrum of the Steklov problem in a domain with a peak

We consider the spectral Steklov problem in a domain with a peak on the boundary. It is shown that the spectrum on the real nonnegative semi-axis can be either discrete or continuous depending on the sharpness of the exponent.

Stability of Cahn-Hilliard fronts

We prove stability of the kink solution of the Cahn-Hilliard equation partial derivative(t)u = partial derivative(x)(2)(-partial derivative(x)(2)u - u/2 + u(3)/2), x is an element of R. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t --> infinity. We prove stability of the kink solution of the Cahn-Hilliard equation partial derivative(t)u = partial derivative(x)(2)(-partial derivative(x)(2)u - u/2 + u(3)/2), x is an element of R. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t --> infinity

Essential spectrum of a periodic elastic waveguide may contain arbitrarily many gaps

We construct a family of periodic elastic waveguides Π h , depending on a small geometrical parameter, with the following property: as h → +0, the number of gaps in the essential spectrum of the elasticity system on Π h grows unboundedly.

Spectra of open waveguides in periodic media

Abstract We study the essential spectra of formally self-adjoint elliptic systems on doubly periodic planar domains perturbed by a semi-infinite periodic row of foreign inclusions. We show that the essential spectrum of the problem consists of the essential spectrum of the purely periodic problem and another component, which is the union of the discrete spectra of model problems in the infinite perturbation strip; these model problems arise by an application of the partial Floquet–Bloch–Gelfand transform.

Toeplitz operators with distributional symbols on Bergman spaces

We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces , 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.

Convergence to a singular steady state of a parabolic equation with gradient blow-up

We study solutions of a parabolic equation which are bounded but whose spatial derivatives blow up in finite time. We establish results on the behavior on the lateral boundary where the singularity occurs and on the rate of convergence to a singular steady state.

Bloch-to-BMOA compositions in several complex variables

We study analytic mappings φ : Bn → Bm and the corresponding analytic composition operators Cφ : f → f ◦ φ. Here n,m ∈ N and Bn is the unit ball of C. In the one complex variable case n = m = 1, D := B1, the investigation of composition operators from the Bloch space B(D) into BMOA(D) has only recently taken place. Boundedness and compactness of Cφ : B(D) → BMOA(D), Cφ : B0(D) → VMOA(D) and Cφ : B(D) → VMOA(D) has been studied in [SZ] by Smith and Zhao and by Makhmutov and Tjani in [MT]. Madigan and Matheson [MM] proved that Cφ is always bounded on B(D). Moreover, [MM] contains a characterization of symbols φ inducing compact composition operators on B(D) and B0(D). The essential norm of a composition operator from B(D) into Qp(D) was computed in [LMT]. In the case of several complex variables, Ramey and Ullrich [RU] have studied the case mentioned in the beginning: their result states that if φ : Bn → D is Lipschitz, then Cφ : B(D) → BMOA(Bn) is well defined, and consequently bounded by the closed graph theorem. Our results below are, of course, more general. The case of Cφ : B(Bn) → B(Bn) was considered by Shi and Luo [SL], where they proved that Cφ is always bounded and gave a necessary and sufficient condition for Cφ to be compact. Our main result states that if φ : Bn → Bm satisfies a very mild regularity condition, then the boundedness of Cφ : B(Bm) → BMOA(Bn) is characterized by the fact that dμφ(z) = (1−|z|2)|Rφ(z)|2 (1−|φ(z)|2)2 dA(z) is a Carleson measure (see notations below). Similarly, a corresponding o–growth condition characterizes the compactness. Let N := {1, 2, 3, . . . }. For z, w ∈ C let 〈z, w〉 = ∑n i=1 ziwi denote the complex inner product on C and |z| = 〈z, z〉1/2. The radial derivative operator is denoted by R; so, if f : Bn → C is analytic, then

Bands in the spectrum of a periodic elastic waveguide

We study the spectral linear elasticity problem in an unbounded periodic waveguide, which consists of a sequence of identical bounded cells connected by thin ligaments of diameter of order